More parents in London and other major local authorities in England have been given their first choice in primary schools this year, despite warnings that the recent mini-boom in the birthrate would lead to increased competition for places.

The Pan London Admissions Board, which co-ordinates school applications in the London boroughs, reported that 81% of applicants received their first preference, an improvement of two percentage points from 2012, when 79% were allocated their first choice.

Oxfordshire, Kent and Essex are among the other local authorities reporting similar results, with Oxfordshire boasting a 91% success rate in placing children in the first choice of school, a five percentage-point improvement over 2012.

A number of London boroughs showed marked improvements on the previous year, including Wandsworth and Westminster, with above-average increases in the numbers getting into their first choice. Overall, 92% of pupils received an offer of a place in one of their first three choices.

Despite the improvement, Helen Jenner, chair of the London admissions board, said: "It is important to emphasise that, however proficient the admission system is – and our arrangements in London are about as efficient and fair as it is possible to have – it cannot create extra places at schools which are already full.

"London local authorities are working hard to try to ensure that they can offer every child a school place for the start of the new academic year, but given the scale of the demand for school places across the capital, this is becoming increasingly difficult."

Kensington and Chelsea remained among the hardest parts of the country for parents to find a place in their top choice for a primary school: just 65% were able to do so this year, little changed from previous years.

London's improvement was aided by a slight fall in the overall number of applications, with the total of 99,000 down by 1% year-on-year. Just 5% of applications received no offers from among their choices.

Roy Leach, Oxfordshire county council's school organisation and planning manager, said: "We are very pleased to be able to offer first-preference places to the overwhelming majority of children. Oxfordshire compares favourably with many other parts of the country in this respect, and a lot of work has taken place in recent years to increase the number of reception places at primary schools across the county."

Oxfordshire county has added 900 additional primary school places in the last five years.

In Essex, 94% of parents received their first choice, while just under 3% missed out on all three choices. Tim Coulson, Essex's director for education, said the rise in pupil numbers resulted in "significant challenges" for the county council. "We are pleased to have worked with local authority-maintained schools and academies to provide 600 more reception class places than last year," he said.

In Kent, 96% of parents received offers from one of their three choices, including 86% who received their first preference. The county has added 450 primary places for the coming school year.

The Department for Education said it was increasing capacity nationally, and allowing the most successful schools to expand.

A spokeswoman for the education department said: "More than £1bn extra has been spent on school places in London in the last year alone. This is on top of the £1.6bn already available to help councils create places where they are most needed. By September, we expect 190,000 extra places will have been created across England, with many more still to come."

The National Audit Office reported last month that more than 240,000 places were needed in primary schools by the 2014 academic year.